Sruthi Reddy

In the number system, irrational numbers are numbers that can’t be written in fractional form. The different types of problems related to irrational numbers are comparison, representation on the number line, rationalizing the denominator and so on. All these types of irrational numbers questions are covered in the Worksheet on Irrational Numbers. So interested students have a look at them and start solving for scoring better marks in the exam.

The problems related to rationalizing the denominator helps to solve the functions of irrational numbers. Given some of the questions are involving calculations of irrational numbers.

More Related Articles:

• Decimal Representation of Irrational Number
• Comparison between Rational and Irrational Numbers
• Problems on Irrational Numbers

Irrational Numbers Worksheet

Question 1:
Check if the below numbers are rational or irrational.
π, \(\frac { 1 }{ √2 } \), \(\frac { √3 }{ 8 } \), √5

Solution:

Since the irrational numbers non-repeating or non-terminating.
So, π, \(\frac { 1 }{ √2 } \), \(\frac { √3 }{ 8 } \), √5 are irrational numbers.

---

Question 2:
Determine whether the following numbers are rational or irrational.
\(\frac { 1 }{ 2 } \), 8, \(\frac { 15 }{ 6 } \), \(\frac { 76 }{ 5 } \)

Solution:

Since the decimal expansion of a rational number either repeats or terminates.
So, \(\frac { 1 }{ 2 } \), 8, \(\frac { 15 }{ 6 } \), \(\frac { 76 }{ 5 } \) are rational numbers.

---

Question 3:
Compare √6 and √5

Solution:

Given two irrational numbers are √6 and √5
We know that if ‘p’ and ‘q’ are two numbers such that ‘q’ is greater than ‘p’, then q² will be greater than p². So, square the given numbers.
√6 = √6 x √6 = (√6)² = 6
√5 = √5 x √5 = (√5)² = 5
6 is greater than 5.
So, √6 is greater than √5.

---

Question 4:
Arrange the following irrational numbers in descending order.
√17, √3, √21, √10, √51

Solution:

Given irrational numbers are √17, √3, √21, √10, √51
We know that if ‘p’ and ‘q’ are two numbers such that ‘q’ is greater than ‘p’, then q² will be greater than p². So, square the given numbers.
√17 = √17 x √17 = (√17)² = 17
√3 = √3 x √3 = (√3)² = 3
√21 = √21 x √21 = (√21)² = 21
√10 = √10 x √10 = (√10)² = 10
√51 = √51 x √51 = (√51)² = 51
Arranging the descending order means placing the numbers from the greatest to the smallest.
So, 51 > 21 > 17 > 10 > 3
Hence, the descending order of numbers is √51 > √21 > √17 > √10 > √3.

---

Question 5:
Write the irrational numbers ∜6, √3 and ∛7 in ascending and descending orders.

Solution:

Given irrational numbers are ∜6, √3 and ∛7
Order of the irrational numbers are 4, 2, 3
The least common multiple of (4, 2, 3) = 12
So, we have to change the order of each number as 12
Change ∜5 as 12th root
∜6 = (4 x 3) √6³
= 12 √216
√3 = (2 x 6) √3⁶
= 12 √729
∛7 = (3 x 4) √7⁴
= 12 √2401
216 < 729 < 2401.
Therefore, ascending order is ∜6, √3, ∛7 and descending order is ∛7, √3 and ∜6.

---

Question 6:
Find an irrational number between √6 and 6.

Solution:

Given two real numbers are √6 and 6
A real number between √6 and 6 is \(\frac { √6 + 6 }{ 2 } \) = ½√6 + 1
But 1 is a rational number and ½√6 is an irrational number. The sum of a rational number and an irrational number is irrational.
So, ½√6 + 1 is an irrational number that lies between √6 and 6.

---

Question 7:
Insert two irrational numbers between √13 and √19.

Solution:

Given two real numbers are √13 and √19
Consider the squares of √13 and √19
(√13)² = 13
(√19)² = 19
Since the numbers 14, 17 lie between 13 and 19 i.e between (√13)² and (√19)²
Therefore, √13 and √19 are √14 and √17.
Hence two irrational numbers between √13 and √19 are √14 and √17.

---

Question 8:
Rationalize \(\frac { 12 }{ 3√10 } \)

Solution:

Given fraction is \(\frac { 12 }{ 3√10 } \)
Since the given fraction has an irrational denominator, so we need to rationalize this and make it more simple. So, to rationalize this, we will multiply the numerator and denominator of the given fraction by root 10, i.e., √10. So,
\(\frac { 12 }{ 3√10 } \) = \(\frac { 12 }{ 3√10 } \) x \(\frac { √10 }{ √10 } \)
= \(\frac { 12√10 }{ 3(10) } \)
= \(\frac { 12√10 }{ 30 } \)
= \(\frac { 2√10 }{ 5 } \)
So, the required rationalized form is \(\frac { 2√10 }{ 5 } \)

---

Question 9:
Rationalize \(\frac { 25 }{ 16 + 2√31 } \)

Solution:

Given fraction is \(\frac { 25 }{ 16 + 2√31 } \)
Multiply numeration, denominator by (16 – 2√31)
\(\frac { 25 }{ 16 + 2√31 } \) = \(\frac { 25 }{ 16 + 2√31 } \) x \(\frac { 16 – 2√31 }{ 16 – 2√31 } \)  [(a + b)(a – b) = a² – b²]
= \(\frac { 25(16 – 2√31) }{ 16² – (2√31)² } \)
= \(\frac { 400 – 50√31 }{ 256 – 124 } \)
= \(\frac { 400 – 50√31 }{ 132 } \)
= \(\frac { 200 – 25√31 }{ 66 } \)
So, the required rationalized form is \(\frac { 200 – 25√31 }{ 66 } \).

---

Question 10:
Rationalize \(\frac { 8 + √6 }{ √5 – 2√7 } \)

Solution:

Given fraction is \(\frac { 8 + √6 }{ √5 + 2√7 } \)
Since, the given problem has an irrational term in the denominator with addition format. So we need to rationalize using the method of multiplication by the conjugate. So,
\(\frac { 8 + √6 }{ √5 + 2√7 } \) = \(\frac { 8 + √6 }{ √5 + 2√7 } \) x \(\frac { √5 – 2√7 }{ √5 – 2√7 } \)
= \(\frac { (8 + √6)(√5 – 2√7) }{ 5 – 4(7) } \)
= \(\frac { 8√5 – 16√7 + √30 – 2√35 }{ 5 – 28 } \)
= \(\frac { 8√5 – 16√7 + √30 – 2√35 }{ -23 } \)
So, the required rationalized form is \(\frac { -8√5 + 16√7 – √30 + 2√35 }{ 23 } \).

---

A matrix is defined as the arrangement of elements in the form of an array. A square matrix is a type of matrix where the number of rows is equal to the number of columns. It is an effective way of arranging elements of the matrix. Learn the definition, concept, examples and operations on a square matrix in the following sections.

Definition of A Square Matrix

A square matrix is a matrix that has an equal number of rows and columns. The n x n matrix is called a square matrix. It is possible to add, subtract and multiply any two square matrices. The multiplication of two square matrices is also a square matrix. So, the number of elements of a square matrix is always a perfect square number.

The important matrices related to a square matrix are as follows:

• Identity Matrix: It is a square matrix that has 1 as the diagonal elements and the remaining elements as zeros.
• Scalar Matrix: It is a square matrix where the diagonal elements are constant numbers and others are equal to zero.
• Trace of a Matrix: The total of the diagonal elements of a matrix is the trace of a matrix.

Also, Check

• Order of a Matrix
• Classification of Matrices
• Row Matrix

Square Matrix Examples

Here we are giving some of the examples of a square matrix with a detailed explanation.

\( A =\left[
\begin{matrix}
11 & 22\cr
33 & 44\cr
\end{matrix}
\right]
\)

The above mentioned matrix is a square matrix of order 2 x 2. The number of rows = 2 = number of columns. So, it is called a square matrix of order 2

\( B =\left[
\begin{matrix}
1 & 2 & 3\cr
4 & 6 & 8\cr
3 & 5 & 7\cr
\end{matrix}
\right]
\)

The above matrix has an order 3 x 3. Since the number of rows and columns are equal, it is a square matrix of order 3. We can find the determinant of the square matrix.

\( C =\left[
\begin{matrix}
14 & 10 & 3 & 11\cr
15 & 1 & 5 & 15\cr
7 & 13 & 2 & 8\cr
6 & 4 & 12 & 16\cr
\end{matrix}
\right]
\)

It is a square matrix of orde 4 x 4.

Operations of Square Matrices

Mathematical operations such as addition, subtraction, multiplication can be performed across two square matrices. Here we are giving the process and examples for a better understanding.

Addition & Subtraction of Two Square Matrices:

Two square matrices can be added/subtracted in a simple way. Let us consider two square matrices.

\(\left[
\begin{matrix}
a1 & a2\cr
a3 & a4\cr
\end{matrix}
\right] \) ±\(\left[
\begin{matrix}
b1 & b2\cr
b3 & b4\cr
\end{matrix}
\right] \) = \(\left[
\begin{matrix}
a1 ± b1 & a2 ± b2\cr
a3 ± b3 & a4 ± b4\cr
\end{matrix}
\right] \)

Examples:

\(\left[
\begin{matrix}
11 & 22\cr
33 & 44\cr
\end{matrix}
\right]
\) + \(\left[
\begin{matrix}
6 & 8\cr
15 & 10\cr
\end{matrix}
\right]
\) = \(\left[
\begin{matrix}
17 & 30\cr
48 & 54\cr
\end{matrix}
\right]
\)

\(\left[
\begin{matrix}
7 & 16\cr
12 & 18\cr
\end{matrix}
\right]
\) – \(\left[
\begin{matrix}
5 & 9\cr
3 & 6\cr
\end{matrix}
\right]
\) = \(\left[
\begin{matrix}
2 & 7\cr
9 & 12\cr
\end{matrix}
\right]
\)

Multiplication of Square Matrix:

Multiplication of a constant with a square is simple.

5A = 5 x \(\left[
\begin{matrix}
2 & 5\cr
10 & 13\cr
\end{matrix}
\right]
\) = \(\left[
\begin{matrix}
10 & 35\cr
45 & 60\cr
\end{matrix}
\right]
\)

The multiplication of two square matrices involves a sequence of steps. We have to multiply the first row of the first matrix with the first column of the second matrix. The following is the detailed process.

\(A = \left[
\begin{matrix}
1 & 3\cr
2 & 4\cr
\end{matrix}
\right]
\) and \(B =\left[
\begin{matrix}
12 & 15\cr
4 & 6\cr
\end{matrix}
\right]
\)

A x B = \(\left[
\begin{matrix}
(1×12 + 2×4) & (1×15 + 3×6)\cr
(2×12 + 4×4) & (2×15 + 4×6)\cr
\end{matrix}
\right]
\) = \(\left[
\begin{matrix}
20 & 33\cr
40 & 54\cr
\end{matrix}
\right]
\)

Transpose of a Square Matrix

The transpose of a matrix is the matrix obtained by transposing the elements of rows into columns, and columns into rows. The order of the given matrix and its transpose is different for some matrices. If the order of a matrix is m x n, then its transpose order is n x m. But in the case of a square matrix, the order is the same.

Example:

\( A =\left[
\begin{matrix}
8 & 10 & 12\cr
14 & 15 & 16\cr
3 & 5 & 7\cr
\end{matrix}
\right]
\)

A^T = \(\left[
\begin{matrix}
8 & 14 & 3\cr
10 & 15 & 5\cr
12 & 16 & 7\cr
\end{matrix}
\right]
\)

In a square matrix, if the given matrix and its transpose are equal, then it is called a symmetric matrix. If the transpose matrix is equal to the negative of the given matrix, then it is a skew-symmetric matrix.

Square Matrix Determinant

The determinant of a matrix is a numerical value or it is a summary value that represents the entire set of elements of the matrix. The determinant of a square matrix having the order 2 x 2 can be easily calculated using the below-given formula.

\( A =\left[
\begin{matrix}
a & b\cr
c & d\cr
\end{matrix}
\right]
\)

|A| = |ad – bc|

If |A| = 0, then it is called a singular matrix otherwise it is a non-singular matrix.

The Inverse of a Square Matrix

The inverse of a matrix is used to divide one matrix with another matrix. You have to calculate the determinant of a square matrix and its adjoint to find its inverse. The inverse of a matrix is obtained by dividing the adjoint matrix with the det of the square matrix.

\( A =\left[
\begin{matrix}
a & b\cr
c & d\cr
\end{matrix}
\right]
\)

A^-1 = \(\frac { 1 }{ |ad – bc| } \) . \(\left[
\begin{matrix}
d & -b\cr
-c & a\cr
\end{matrix}
\right]
\) = \(\frac { 1 }{ |A| } \) . adj(A)

A square matrix is called an orthogonal matrix if A ^T = A^-1

Important Properties of Square Matrix

The important properties of a square matrix are given here:

• The number of rows is equal to the number of columns.
• The sum of the elements of a square matrix is the trace of a matrix.
• The order of the transpose and original matrices are the same.
• We can perform different operations like addition, subtraction, multiplication and inverse on a square matrix.
• The determinant also can be calculated easily.

Example Questions & Answers

Question 1:
Find the transpose of the square matrix \( A =\left[
\begin{matrix}
4 & 9\cr
11 & 35\cr
\end{matrix}
\right]
\)

Answer:
Given matrix is \( A =\left[
\begin{matrix}
4 & 9\cr
11 & 35\cr
\end{matrix}
\right]
\)
It’s transpose is \(\left[
\begin{matrix}
4 & 11\cr
9 & 35\cr
\end{matrix}
\right]
\)

Question 2:
Find the multiplication of two square matrices \( A =\left[
\begin{matrix}
1 & 8\cr
7 & 5\cr
\end{matrix}
\right]
\) and \( B =\left[
\begin{matrix}
5 & 4\cr
6 & 7\cr
\end{matrix}
\right]
\)

Answer:
The given matrices are \( A =\left[
\begin{matrix}
1 & 8\cr
7 & 5\cr
\end{matrix}
\right]
\) and \( B =\left[
\begin{matrix}
5 & 4\cr
6 & 7\cr
\end{matrix}
\right]
\)
A x B = \(\left[
\begin{matrix}
1×5 + 8×6 & 1×4 + 8×7\cr
7×5 + 5×6 & 7×4 + 5×7\cr
\end{matrix}
\right]
\) = \(\left[
\begin{matrix}
53 & 60\cr
65 & 63\cr
\end{matrix}
\right]
\)

Question 3:
Find the inverse of the given square matrix \(C = \left[
\begin{matrix}
5 & 6\cr
2 & 3\cr
\end{matrix}
\right]
\)

Answer:
Given matrix is \(C = \left[
\begin{matrix}
5 & 6\cr
2 & 3\cr
\end{matrix}
\right]
\)
|C| = |ad – bc| = |15 – 12| = 3
C^-1 = \(\frac { 1 }{ 3 } \) . \( \left[
\begin{matrix}
3 & -6\cr
-2 & 5\cr
\end{matrix}
\right]
\) = \( \left[
\begin{matrix}
1 & -2\cr
-2/3 & 5/3\cr
\end{matrix}
\right]
\)

FAQ’s on Square Matrix

1. What is a square matrix give an example?

A square matrix is a m x m matrix, where first m is the number of rows and second m is the number of columns. The example is \( A =\left[
\begin{matrix}
3 & 4 & 9\cr
12 & 11 & 35\cr
12 & 11 & 35\cr
\end{matrix}
\right]
\)

2. How to find a square matrix?

A square matrix can be found by checking the number of rows and columns in it. If they are equal, then it is said to be a square matrix.

3. What is squaring a matrix?

Squaring a matrix means multiplying a matrix by its own. Before multiplying the matrices, you have to check whether the number of columns of the first matrix is equal to the number of rows of the second matrix or not.

4. What are the dimensions of a square matrix?

The dimension of a square matrix is the number of rows by the number of columns. Here, both numbers should be equal.

A matrix is a rectangular arrangement of elements i.e numbers into m rows and n columns. The order of that particular matrix is defined as m x n. Row matrix is a type of matrix where the number of rows is always equal to one. Go through the complete article to know the useful details like definition, examples, properties and operations of row matrix.

What is a Row Matrix?

A row matrix is a matrix that has only one row. A matrix of an order m x n, where m is the number of rows, n is the number of columns is said to be a row matrix if and only if, m = 1. Mathematically, a row matrix can be expressed as \( A =\left[
\begin{matrix}
a11 & a12 & a13 & . . . & a1n\cr
\end{matrix}
\right]
\)

The order of the row matrix is 1 x n and n is the number of elements in it. It is not a square matrix so it is not possible to find the determinant of it. The horizontal lines of elements form a row matrix. Read the below sections to know more important details about the single row matrix.

Examples of Row Matrix

Some of the examples of the row matrix are given here.

\(\left[
\begin{matrix}
5\cr
\end{matrix}
\right]
\)

• The order of above matrix is 1 x 1

\(\left[
\begin{matrix}
1 & 6\cr
\end{matrix}
\right]
\)

• The order of above matrix is 1 x 2

\(\left[
\begin{matrix}
7 & 2 & 3\cr
\end{matrix}
\right]
\)

• The order of above matrix is 1 x 3

\(\left[
\begin{matrix}
10 & 9 & 8 & 7\cr
\end{matrix}
\right]
\)

• The order of above matrix is 1 x 4

\(\left[
\begin{matrix}
10 & 20 & 30 & 40 &50\cr
\end{matrix}
\right]
\)

• The order of above matrix is 1 x 5.

Row Matrix – Properties

Below given properties of the row matrix are helpful for a better understanding of this matrix.

• It has only one row.
• It can have any number of columns.
• It is also a rectangular matrix.
• The number of elements in a row matrix is equal to the number of columns in it.
• The transpose of a row matrix is a column matrix.
• A row matrix can be multiplied only by a column matrix.
• The row matrix can be added to or subtracted from a row matrix of the same order.
• The product of a row matrix and a column matrix gives the singleton matrix as the product.

Operations on Row Matrix

The algebraic operations such as addition, subtraction, multiplication can be performed on two or more row matrices.

Row Matrices Addition and Subtraction:

To perform an addition or subtraction operation, two matrices must have the same order. The elements of the first matrix are added to or subtracted from the respective elements of the second matrix in case of addition or subtraction.

Examples:

\( A =\left[
\begin{matrix}
12 & 11 & 35\cr
\end{matrix}
\right]
\), \( B =\left[
\begin{matrix}
3 & 4 & 9\cr
\end{matrix}
\right]
\)

\(A + B =\left[
\begin{matrix}
12 + 3 & 11 + 4 & 35 + 9\cr
\end{matrix}
\right]
\) = \(\left[
\begin{matrix}
15 & 15 & 44\cr
\end{matrix}
\right]
\)

\(A – B =\left[
\begin{matrix}
12 – 3 & 11 – 4 & 35 – 9\cr
\end{matrix}
\right]
\) = \(\left[
\begin{matrix}
9 & 7 & 26\cr
\end{matrix}
\right]
\)

Multiplication of Row Matrix:

Multiplication of row matrices is possible only with a column matrix and the product is a singleton matrix.

Example:

\( A =\left[
\begin{matrix}
6 & 5 & 1\cr
\end{matrix}
\right]
\), \( B =\left[
\begin{matrix}
3\cr
4\cr
2\cr
\end{matrix}
\right]
\)

\(  A x B =\left[
\begin{matrix}
3 x 6 + 4 x 5 + 1 x 2\cr
\end{matrix}
\right]
\) = \(\left[
\begin{matrix}
40\cr
\end{matrix}
\right]
\)

Questions on Row Matrix

Question 1:
Find the size of the matrix \( A =\left[
\begin{matrix}
2 & 1 & 5 & 6 & 8 & 13 & 42\cr
\end{matrix}
\right]
\)

Solution:
The order of the matrix \( A =\left[
\begin{matrix}
2 & 1 & 5 & 6 & 8 & 13 & 42\cr
\end{matrix}
\right]
\) is 1 x 7 and it has 7 elements in it.

Question 2:
Find the transpose of \(\left[
\begin{matrix}
5 & 6 & 1 & 8 & 10\cr
\end{matrix}
\right]
\)

Solution:
The transpose of a row matrix is a column matrix.
The transpose of a given matrix is \(\left[
\begin{matrix}
5\cr
6\cr
1\cr
8\cr
10\cr
\end{matrix}
\right]
\)

Question 3:
Get the sum and difference of the following matrices.
\( A =\left[
\begin{matrix}
9 & 7 & 3 & 2 & 2 & 1 & 6\cr
\end{matrix}
\right]
\), \( B =\left[
\begin{matrix}
1 & 7 & 10 & 5 & 2 & 3 & 4\cr
\end{matrix}
\right]
\)

Solution:
Given matrices are \( A =\left[
\begin{matrix}
9 & 7 & 3 & 2 & 2 & 1 & 6\cr
\end{matrix}
\right]
\), \( B =\left[
\begin{matrix}
1 & 7 & 10 & 5 & 2 & 3 & 4\cr
\end{matrix}
\right]
\)
\( A + B =\left[
\begin{matrix}
9 + 1 & 7 + 7 & 3 + 10 & 2 + 5 & 2 + 2 & 1 + 3 & 6 + 4\cr
\end{matrix}
\right]
\) = \(\left[
\begin{matrix}
10 & 14 & 13 & 7 & 4 & 4 & 10\cr
\end{matrix}
\right]
\)
\( A – B =\left[
\begin{matrix}
9 – 1 & 7 – 7 & 3 – 10 & 2 – 5 & 2 – 2 & 1 – 3 & 6 – 4\cr
\end{matrix}
\right]
\) = \(\left[
\begin{matrix}
8 & 0 & -7 & -3 & 0 & -2 & 2\cr
\end{matrix}
\right]
\)

Frequently Asked Questions

1. What is row matrix and example?

A row matrix is a matrix that has only one row. The example is \(\left[
\begin{matrix}
5 & 8 & 15 & 10\cr
\end{matrix}
\right]
\).

2. What is row matrix order?

The row matrix order depends on the number of columns or elements it has because the number of rows is always equal to 1. The general row matrix order is 1 x n. Where n is the number of elements in a row matrix.

3. What is the difference between row and column matrices?

In a row matrix, the elements are arranged in a horizontal manner. The column matrix has elements arranged in a vertical manner. The order of a column matrix is n x 1 and the row matrix is 1 x n. The product of a row and column matrix results in a singleton matrix.

4. What is the transpose of a row matrix?

The transpose of a row matrix is a column matrix. The row matrix of order 1 x n is transposed into a column matrix of order n x 1.

The rectangular arrangement of an array of numbers is called a matrix. Matrices are the plural form of the matrix. These matrices are classified into various types depending on the number of elements present in them, number of rows, columns, order, size and so on. Some of the matrices types are row matrix, column matrix, null matrix, identity matrix, singleton matrix, and others.

Here we are giving the example questions and answers of classification of matrices in the following sections. Students can check those practice questions and solve them to get a good score in the exam.

Also, check

• Definition of a Matrix
• Classification of Matrices
• Position of an Element in a Matrix
• Oder of a Matrix

Solved Problems on Classification of Matrices

Problem 1:
Let \( A =\left[
\begin{matrix}
3 & 4 & 9\cr
0 & 1 & 3\cr
\end{matrix}
\right]
\), \( B =\left[
\begin{matrix}
0 & 0 & 0\cr
0 & 0 & 0\cr
0 & 0 & 0\cr
\end{matrix}
\right]
\), \( C =\left[
\begin{matrix}
8 & 9 & 5\cr
6 & 0 & 4\cr
1 & 5 & 2\cr
\end{matrix}
\right]
\), \( D =\left[
\begin{matrix}
14 & 8 & 5\cr
\end{matrix}
\right]
\), \( E =\left[
\begin{matrix}
100\cr
\end{matrix}
\right]
\)
Identify the type of each matrix.

Solution:
\( A =\left[
\begin{matrix}
3 & 4 & 9\cr
0 & 1 & 3\cr
\end{matrix}
\right]
\)
Matrix A is a rectangular matrix. As the number of rows is 2 which is not equal to the number of columns 3.
\( B =\left[
\begin{matrix}
0 & 0 & 0\cr
0 & 0 & 0\cr
0 & 0 & 0\cr
\end{matrix}
\right]
\)
Matrix B is a Null matrix as the all its elements are zero’s.
\( C =\left[
\begin{matrix}
8 & 9 & 5\cr
6 & 0 & 4\cr
1 & 5 & 2\cr
\end{matrix}
\right]
\)
Matrix C is a square matrix. As the number of rows = number of columns i.e 3 = 3.
\( D =\left[
\begin{matrix}
14 & 8 & 5\cr
\end{matrix}
\right]
\)
Matrix D is a row matrix as it has only one row.
\( E =\left[
\begin{matrix}
100\cr
\end{matrix}
\right]
\)
Matrix E is a singleton matrix as it has only one element.

Problem 2:
(i) Construct a 2 x 3 matrix that has elements as natural numbers from 10 to 16.
(ii) Construct a 2 x 2 simple identity matrix.

Solution:
(i) Matrix that has numbers from 10 to 16 is \( \left[
\begin{matrix}
10 & 11 & 12\cr
13 & 14 & 15\cr
\end{matrix}
\right]
\)
(ii) 2 x 2 Identity matrix is \( \left[
\begin{matrix}
1 & 0\cr
0 & 1\cr
\end{matrix}
\right]
\)

Problem 3:
Identify which of the following is a diagonal matrix.
\( A =\left[
\begin{matrix}
3 & 4 & 9\cr
12 & 11 & 35\cr
\end{matrix}
\right]
\), \( B =\left[
\begin{matrix}
5 & 0 & 0\cr
0 & 5 & 0\cr
\end{matrix}
\right]
\), \( C =\left[
\begin{matrix}
3 & 0\cr
0 & 1\cr
\end{matrix}
\right]
\)

Solution:
\( C =\left[
\begin{matrix}
3 & 0\cr
0 & 1\cr
\end{matrix}
\right]
\) is a diagonal matrix as its diagonal elements are non-zero and the remaining elements are zeros.
The remaining matrices don’t satisfy the diagonal matrix condition.

Problem 4:
\( A =\left[
\begin{matrix}
1 & 6 & 7\cr
2 & 4 & 6\cr
3 & 2 & 5\cr
\end{matrix}
\right]
\), \( B =\left[
\begin{matrix}
0 & 0 & 0\cr
0 & 0 & 0\cr
0 & 0 & 0\cr
\end{matrix}
\right]
\), \( C =\left[
\begin{matrix}
18 & 0\cr
0 & 18\cr
\end{matrix}
\right]
\), \( D =\left[
\begin{matrix}
12 & 10 & 0 & 2\cr
8 & 5 & 4 & 6\cr
\end{matrix}
\right]
\), \( E =\left[
\begin{matrix}
4 & 0 & 0\cr
2 & 5 & 0\cr
1 & 3 & 6\cr
\end{matrix}
\right]
\), \( F =\left[
\begin{matrix}
180 & 0 & 0\cr
0 & 120 & 0\cr
0 & 0 & 100\cr
\end{matrix}
\right]
\)
Identify the following matrices.
(i) Horizontal matrix
(ii) Scalar matrix
(iii) Null matrix
(iv) Square Matrix
(v) Triangular Matrix
(vi) Diagonal Matrix

Solution:
(i) D is the horizontal matrix as its number of rows are less than the number of columns.
(ii) C is a scalar matrix. Here the diagonal elements are the same and the remaining elements are zeros.
(iii) B is a null matrix as all its elements are zeros.
(iv) A is a square matrix
(v) E is a lower triangular matrix
(vi) F is a diagonal matrix.

Problem 5:
Give a 4 x 4 order example for the below-mentioned matrices.
(i) Symmetric Matrix
(ii) Upper Triangular Matrix
(iii) Boolean Matrix
(iv) Matrix of Ones

Solution:
(i) \( X =\left[
\begin{matrix}
3 & 6 & 9 & 12\cr
6 & 12 & 18 & 24\cr
9 & 18 & 27 & 36\cr
12 & 24 & 36 & 42\cr
\end{matrix}
\right]
\) is a symmetric matrix.
(ii) \( Y =\left[
\begin{matrix}
1 & 2 & 3 & 5\cr
0 & 4 & 6 & 7\cr
0 & 0 & 8 & 9\cr
0 & 0 & 0 & 11\cr
\end{matrix}
\right]
\) is an upper triangular matrix.
(iii) \( Z =\left[
\begin{matrix}
1 & 0 & 0 & 1\cr
0 & 1 & 1 & 0\cr
1 & 0 & 0 & 1\cr
0 & 0 & 1 & 1\cr
\end{matrix}
\right]
\)
(iv) \( S =\left[
\begin{matrix}
1 & 1 & 1 & 1\cr
1 & 1 & 1 & 1\cr
1 & 1 & 1 & 1\cr
1 & 1 & 1 & 1\cr
\end{matrix}
\right]
\) is a matrix of ones.

A matrix is a rectangular array of numbers. The numbers in a matrix are enclosed by [] or (). The size of a matrix is defined by the number of rows and columns in it. The matrices are classified into different types on the basis of the value of their element, size, number of rows, columns, etc. Learn the definitions, examples of matrices types in the following sections.

Classification of Matrices

The matrices are classified in various based depending on their size, order of a matrix, number of rows, no of columns, elements, the position of an element in matrix and others. Here the list of classification of matrices is provided.

• Row Matrix
• Column Matrix
• Zero or Null Matrix
• Singleton Matrix
• Horizontal Matrix
• Vertical Matrix
• Square Matrix
• Diagonal Matrix
• Scalar Matrix
• Identity (Unit) Matrix
• Equal Matrix
• Triangular Matrices
• Singular Matrix
• Non-Singular Matrix
• Symmetric Matrices
• Skew-Symmetric Matrices
• Hermitian Matrix
• Skew – Hermitian Matrix
• Orthogonal Matrix
• Idempotent Matrix
• Involuntary Matrix
• Nilpotent Matrix

Types of Matrices Based on Dimension

The detailed explanations of types of matrices based on their dimensioned are included here:

Row Matrix:

A matrix that has exactly only one row is called a row matrix.

Example:

\( A =\left[
\begin{matrix}
8 & 5 & 2\cr
\end{matrix}
\right]
\)

Column Matrix:

A matrix that has only one column is called a column matrix.

Example:

\( B =\left[
\begin{matrix}
3\cr
1\cr
7\cr
\end{matrix}
\right]
\)

Square Matrix:

Any matrix in the order of m x n and where m = n is called a square matrix or a matrix that has an equal number of rows and columns in it is a square matrix.

Example:

\( C =\left[
\begin{matrix}
1 & 2 & 3\cr
9 & 8 & 7\cr
4 & 5 & 6\cr
\end{matrix}
\right]
\)

Rectangular Matrix:

A matrix of the order m x n and in which m ≠ n is called a rectangular matrix.

Example:

\( D =\left[
\begin{matrix}
4 & 2 & 8\cr
6 & 1 & 5\cr
\end{matrix}
\right]
\)

Singleton Matrix:

A matrix that has only one element is called a singleton matrix. Its order is 1 x 1.

Examples:

\( M =\left[
\begin{matrix}
3 \cr
\end{matrix}
\right]
\), \( N =\left[
\begin{matrix}
12 \cr
\end{matrix}
\right]
\)

Horizontal Matrix:

A matrix in which the number of columns is more than the number of rows is called a horizontal matrix.

Example:

\( A =\left[
\begin{matrix}
1 & 5 & 9\cr
13 & 18 & 25 \cr
\end{matrix}
\right]
\)

Vertical Matrix:

A matrix in which the number of rows is greater than the number of columns is called a vertical matrix.

Example:

\( C =\left[
\begin{matrix}
2 & 4\cr
3 & 6\cr
4 & 8\cr
1 & 2\cr
\end{matrix}
\right]
\)

Diagonal Matrix:

A square matrix where all the elements, except the principal diagonal, are zero is called a diagonal matrix.

Example:

\( D =\left[
\begin{matrix}
30 & 0 & 0\cr
0 & 6 & 0\cr
0 & 0 & 7\cr
\end{matrix}
\right]
\)

Scalar Matrix:

If all the elements in the diagonal of a diagonal matrix are equal, it is called a scalar matrix.

Example:

\( X =\left[
\begin{matrix}
5 & 0 & 0\cr
0 & 5 & 0\cr
0 & 0 & 5\cr
\end{matrix}
\right]
\), \( Y =\left[
\begin{matrix}
8 & 0\cr
0 & 8\cr
\end{matrix}
\right]
\)

Identity Matrix:

It is a square matrix in which all elements in the leading diagonal are 1 and the remaining elements are zeros. It is also called a unit matrix.

Examples:

\( E =\left[
\begin{matrix}
1 & 0 & 0\cr
0 & 1 & 0\cr
0 & 0 & 1\cr
\end{matrix}
\right]
\) and \( F =\left[
\begin{matrix}
1 & 0\cr
0 & 1\cr
\end{matrix}
\right]
\)

Matrix of Ones:

A matrix in which all the elements are 1, then it is called a matrix of ones.

Example:

\( G =\left[
\begin{matrix}
1 & 1 & 1\cr
1 & 1 & 1\cr
1 & 1 & 1\cr
\end{matrix}
\right]
\)

Zero Matrix:

A matrix in which all the elements are zero is called a zero matrix or null matrix.

Example:

\( H =\left[
\begin{matrix}
0 & 0 & 0\cr
0 & 0 & 0\cr
0 & 0 & 0\cr
\end{matrix}
\right]
\)

Explanations on Classification of Matrices

Triangular Matrix:

If the elements above or below the principal diagonal of a square matrix are zero, then it is a triangular matrix.

Examples:

\( D =\left[
\begin{matrix}
3 & 13 & 15\cr
0 & 6 & 8\cr
0 & 0 & 7\cr
\end{matrix}
\right]
\) and \( C =\left[
\begin{matrix}
8 & 0 & 0\cr
10 & 7 & 0\cr
11 & 15 & 4\cr
\end{matrix}
\right]
\)

Symmetric Matrix:

A square matrix is said to be symmetric matrix if the original matrix is equal to the transpose of the given matrix.

Example:

\( B =\left[
\begin{matrix}
8 & 10 & 6\cr
10 & 7 & 15\cr
6 & 15 & 4\cr
\end{matrix}
\right]
\)
and \( B^T =\left[
\begin{matrix}
8 & 10 & 6\cr
10 & 7 & 15\cr
6 & 15 & 4\cr
\end{matrix}
\right]
\)

Here, B = B^T;. So, these are symmetric matrices.

Skew-symmetric Matrix:

A square matrix is said to be skew-symmetric matrix if the original matrix is equal to the negative of the transpose of the given matrix.

Example:

\( F =\left[
\begin{matrix}
0 & 3\cr
-3 & 0\cr
\end{matrix}
\right]
\) \( F^T =\left[
\begin{matrix}
0 & -3\cr
3 & 0\cr
\end{matrix}
\right]
\) \( -F =\left[
\begin{matrix}
0 & -3\cr
3 & 0\cr
\end{matrix}
\right]
\)

So, F^T = -F.

Questions on Types of Matrices

Question 1:
Identify the class of each matrix.
\( M =\left[
\begin{matrix}
3 & 4 & 9\cr
\end{matrix}
\right]
\), \( N =\left[
\begin{matrix}
3 & 4 & 9\cr
12 & 11 & 35 \cr
3 & 4 & 9\cr
\end{matrix}
\right]
\), \( O =\left[
\begin{matrix}
3 & 4\cr
12 & 11\cr
\end{matrix}
\right]
\)

Solution:
\( M =\left[
\begin{matrix}
3 & 4 & 9\cr
\end{matrix}
\right]
\)
M is a row matrix as it has exactly one row.
\( N =\left[
\begin{matrix}
3 & 4 & 9\cr
12 & 11 & 35 \cr
3 & 4 & 9\cr
\end{matrix}
\right]
\)
N is a square matrix as it has an equal number of rows and columns.
\( O =\left[
\begin{matrix}
3 & 4\cr
12 & 11\cr
\end{matrix}
\right]
\)
O is a square matrix. Because it has 2 rows, 2 columns.

Question 2:
Construct a null matrix of order 4 x 3 and unit matrix of order 4 x 4.

Solution:
A null matrix of order 4 x 3 is \( A =\left[
\begin{matrix}
0 & 0 & 0\cr
0 & 0 & 0\cr
0 & 0 & 0\cr
0 & 0 & 0\cr
\end{matrix}
\right]
\)
Identity matrix of order 4 x 4 is \( A =\left[
\begin{matrix}
1 & 0 & 0 & 0\cr
0 & 1 & 0 & 0\cr
0 & 0 & 1 & 0\cr
0 & 0 & 0 & 1\cr
\end{matrix}
\right]
\)

FAQ’s on Matrices Classification

1. What are the four types of the matrix?

The 4 different types of matrices are row matrix, column matrix, null matrix, and square matrix.

2. How many types of matrices are there?

On average, 22 types of matrices are there.

3. What are the properties of matrices?

The important properties of matrices are associative property, distributive property, identity property, property of zero, closure property and so on.

4. How to classify matrices?

Basically, matrices are classified as per the number of rows, number of columns, specific elements in them.

A rectangular array of m x n numbers in the form of rows and columns is called a matrix. The numbers are enclosed by [] or () symbols. Here we will learn the definitions, examples, properties, and types of matrics. Also, check the solved example questions and more useful details about the matrix in the following sections.

Definition of a Matrix

A matrix is defined as the rectangular arrangement or array of numbers or functions in the form of horizontal lines (rows) and vertical lines (columns) and is subject to certain rules of operation. Matrices are denoted by the capital letters of the alphabet.

Example:

The example of a matrix is given here.
\( A =\left[
\begin{matrix}
3 & 4 & 9\cr
12 & 11 & 35 \cr
\end{matrix}
\right]
\)

In the above matrix A, the numbers 3, 4, 9, 12, 11, and 35 are the elements of the matrix. The number 3 belongs to the 1st row and 1st column so it is called (1, 1)th element of the matrix A. As matrix A has 2 rows and 3 columns, the order of the matrix is 2 x 3. other important topics on matrices are provided below.

• Position of an Element in a Matrix
• Classification of Matrices
• Problems on Classification of Matrices
• Square Matrix
• Row Matrix
• Column Matrix
• Null Matrix
• Equal Matrices
• Identity (or Unit) Matrix
• Triangular Matrix
• Addition of Matrices
• Addition of Two Matrices
• Properties of Addition of Matrices
• Negative of a Matrix
• Subtraction of Matrices
• Subtraction of Two Matrices
• Scalar Multiplication of a Matrix
• Multiplication of a Matrix by a Number
• Properties of Scalar Multiplication of a Matrix
• Multiplication of Matrices
• Multiplication of Two Matrices
• Problems on Understanding Matrices
• Worksheet on Understanding Matrix
• Worksheet on Addition of Matrices
• Worksheet on Matrix Multiplication
• Worksheet on Matrix

Types of Matrices

The different types of matrices and their definitions are given here.

• Symmetric Matrix: A square matrix A = [a_ij] is called a symmetric matrix if a_ij = a_ji, for all i, j.
• Skew-Symmetric Matrix: A matrix A = [a_ij] is called skew-symmetric matrix is a_ij = -a_ji.
• Orthogonal matrix: If AA^T = In = A^TA
• Hermitian and skew-Hermitian matrix: A = A^θ (A^θ represent conjugate transpose). A^θ = -A is called skew-Hermitian matrix.
• Idempotent Matrix: If A² = A
• Involuntary materix: If A² = I or A^-1 = A
• Nilpotent Matrix: A square matrix A is nilpotebt, if A^p = 0, p is an interger.

Operations on Matrices

Matrix operations involve three algebra operations which are the addition of matrices, subtraction of matrices and multiplication of matrices. The details of each are given here.

Matrix Addition:

If A[a_ij]_mxn and B[b_ij]_mxn are two matrices having the same order, then their sim is A + B also a matrix. Every element of matrix A are added to the corresponding element of matrix B and the sum matrix will also have the same order.

The sum of 2 x 2 matrices is given as:

\( \left[
\begin{matrix}
a1 & a2 \cr
a3 & a4 \cr
\end{matrix}
\right]
\) + \( \left[
\begin{matrix}
b1 & b2 \cr
b3 & b4 \cr
\end{matrix}
\right]
\) = \( \left[
\begin{matrix}
a1 + b1 & a2 + b2 \cr
a3 + b3 & a4 + b4\cr
\end{matrix}
\right]
\)

Subtraction of Matrices:

If A and B are any two matrices having the same order, then A – B = A + (-B). The difference of two 2 x 2 matrices is given as:

\( \left[
\begin{matrix}
a1 & a2 \cr
a3 & a4 \cr
\end{matrix}
\right]
\) – \( \left[
\begin{matrix}
b1 & b2 \cr
b3 & b4 \cr
\end{matrix}
\right]
\) = \( \left[
\begin{matrix}
a1 – b1 & a2 – b2 \cr
a3 – b3 & a4 – b4\cr
\end{matrix}
\right]
\)

The matrix subtraction involves subtracting each element of one matrix from the corresponding element of the second matrix. The obtained matrix also have the same order.

Matrix Multiplication:

If A and B be any two matrices, then their product AB will be defined only when the number of columns in A is equal to the number of rows in B. If A = [a_ij]_mxn and B = [b_ij]_nxp then their product AB = C = [c_ij]_mxp and (AB)_ij = C_ij = ∑_r=1ⁿ a_irb_rj.

Properties of Matrix

Matrix properties for addition operation are provided here. You can check the properties of matrices page to know other properties.

1. Commutative Law:

If A and B are two matrices, then A + B = B + A. For three matrices A, B, C, commutative law is if A + B = A + C then B + A = C + A and B = C.

2. Associative Law:

For any three matrices A, B, C the asssociative law states that (A + B) + C = A + (B + C).

3. Identity Law:

A matrix is added to the given matrix, and the sum is the same as the given matrix, called the identity matrix for addition.

A + O = O + A = A

Here O is the zero matrix.

4. Additive Inverse:

A matrix added to the given matrix to get the sum zero is called additive inverse.

A + (-A) = 0 = (-A) + A

Examples on Matrix

Example 1:
If \( B =\left[
\begin{matrix}
8 & 5 & 4\cr
6 & 8 & 2 \cr
\end{matrix}
\right]
\). What is the order of matrix B?

Answer:
The order of matrix B is 2 × 3 because there are 2 rows and 3 columns in the matrix.

Example 2:
If a matrix has 8 elements, find the possible orders of the matrix.

Answer:
8 = 1 x 8
8 = 8 x 1
8 = 4 x 2
8 = 2 x 4
Therefore, the possible orders of the matrix are 1 x 8, 8 x 1, 4 x 2, 2 x 4.

Frequently Asked Question’s on Matrix

1. What is the best definition of a matrix?

A matrix is a collection of numbers arranged into a fixed number of rows and columns. The numbers are real numbers. The numbers are enclosed by [] or () parenthesis.

2. What is the matrix in real life?

In real life, matrices are applied in the study of electrical circuits, quantum mechanics, optics. It is also helpful for the calculation of battery power outputs.

3. What is another word for the matrix?

Another word that defines a matrix is Array, grid, table or spreadsheet.

Probability means possibility. It deals with the occurrence of a random event. The example is if you toss a coin, the result will be head or tail. In this case, we use the probability method. Check the following sections to know what is the probability of rolling three dice, solved questions.

Probability for Rolling Three Dice

To find the probability of any event, we have to know the number of favorable outcomes and the total number of outcomes. If you throw a die, the sample space contains numbers from 1 to 6 and the total number of outcomes is 6. In the same way, when three dice are rolled simultaneously the probability becomes difficult. The total number of outcomes in rolling 3 dice is 6³ = 216.

The faces of a die are {1, 2, 3, 4, 5, 6}. You can check the probabilities of rolling three dice example questions with solutions.

More Related Articles:

• Probability of Tossing Two Coins
• Probability for Rolling Two Dice
• Events in Probability

Worked-out problems on 3 Dice Rolling Probability

Problem 1:
Three dice are rolled. What is the probability that the numbers shown are different?

Solution:
The total number of outcomes = 6 x 6 x 6 = 216
Three dice shows different numbers means, if the first dice shows 2, 2nd dice shows 3, 3rd dice shows 4.
The total number of favorable outcomes = 6 x 5 x 4 = 120
P(numbers shown are different) = \(\frac { Number of favorable outcomes }{ Total number of outcomes } \)
= \(\frac { 120 }{ 216 } \)
= \(\frac { 5 }{ 9 } \)
Therefore, the probability that the numbers shown are different is \(\frac { 5 }{ 9 } \).

Problem 2:
Three dice are thrown together. Find the probability of
(i) getting a sum of 6
(ii) getiing a total of 5
(iii) getting a total of atmost of 6

Solution:
Three different dice are thrown at the same time.
Total no of possible outcomes = 6³ = 216
(i) getting a sum of 6
No of events of getting a total of 6 = 10
Possibilities of getting a sum of 6 = {(1, 1, 4), (1, 4, 1), (4, 1, 1), (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1) and (2, 2, 2)}
P(getting a sum of 6) = \(\frac { Number of favorable outcomes }{ Total number of outcomes } \)
= \(\frac { 10 }{ 216 } \)
= \(\frac { 5 }{ 108 } \)
(ii) getiing a total of 5
No of events of getting a total of 5 = 6
Possibilities of getting a sum of 5 = {(1, 1, 3), (1, 3, 1), (3, 1, 1), (2, 2, 1), (2, 1, 2) and (1, 2, 2)}
P(getting a total of 5) = \(\frac { Number of favorable outcomes }{ Total number of outcomes } \)
= \(\frac { 6 }{ 216 } \)
= \(\frac { 1 }{ 36 } \)
(iii) getting a total of atmost of 6
No of events of getting a total of atmost 6 = 20
Possibilities of getting a total of atmost 6 = {(1, 1, 1), (1, 1, 2), (1, 2, 1), (2, 1, 1), (1, 1, 3), (1, 3, 1), (3, 1, 1), (2, 2, 1), (1, 2, 2), (1, 1, 4), (1, 4, 1), (4, 1, 1), (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1) and (2, 2, 2)}
P(getting a total of atmost 6) = \(\frac { Number of favorable outcomes }{ Total number of outcomes } \)
= \(\frac { 20 }{ 216 } \)
= \(\frac { 5 }{ 54 } \)

FAQ’s on Probability of Rolling Three Dice

1. What is the probability of getting 3 sixes when you roll 3 dice?

The number of possible outcomes is 216 when you roll 3 dice.

The number of possibilities to get 3 sixes is 1

Therefore, P(getting 3 sixes) = \(\frac { 1 }{ 216 } \)

2. How to find the probability of rolling multiple dice?

The simple formula to get the probability of rolling multiple dice are \(\frac { number of desired outcomes }{ number of possible outcomes } \).

3. What is the probability of getting at least one six when you roll 3 dice?

The number of possible outcomes = 216

The number of possibilities of getting at least one six = 5 x 5 x 5 = 125

P(getting at least one six) = \(\frac { 125 }{ 216 } \).

The probability is the chance of the occurrence of an event. For example, the chance of getting a head while tossing a coin is ½. The value of probability always lies between 0 and 1. In general, the questions in probability are related to rolling a die, tossing a coin, choosing a card from a pack of cards and so on.

The formula of probability is the ratio of the number of favorable outcomes to the total number of outcomes. The terms that define probability are events, outcome, sample space, etc. Students can get the solved problems on probability in the following sections.

Also, Check

• Coin Toss Probability
• Random Experiments
• Probability and Playing Cards

Solved Probability Problems

Problem 1:
A bag contains 8 balls numbered 1 to 8
(i) What is the probability of selecting 1 from the bag?
(ii) What is the probability of selecting an odd number?
(iii) What is the probability of selecting a number less than 4?

Solution:
Total number of bags = 8
Sample sapce S = {1, 2, 3, 4, 5, 6, 7, 8}
(i)
Number of 1 balls = 1
Probability of selecting 1 from the ball = \(\frac { Number of favorable outcomes }{ Total number of outcomes } \)
= \(\frac { 1 }{ 8 } \)
(ii)
Number of odd number balls = 5
Probability of selecting odd numbered ball = \(\frac { Number of favorable outcomes }{ Total number of outcomes } \)
= \(\frac { 5 }{ 8 } \)
(iii)
Number of balls which are less than 4 = 3
Probability of selecting odd numbered ball = \(\frac { Number of favorable outcomes }{ Total number of outcomes } \)
= \(\frac { 3 }{ 8 } \)

Problem 2:
Two coins are tossed simultaneously, find the probability that two heads are obtained.

Solution:
Sample space S = {HT, TH, HH, TT}
Let E be the event of obtaining two heads
So, E = {HH}
Probability of getting two heads = \(\frac { Number of favorable outcomes }{ Total number of outcomes } \)
= \(\frac { 1 }{ 4 } \)

Problem 3:
A single card is drawn from a pack of 52 cards. Find the probability of
(a) The card is not king
(b) The card is a red king
(c) The card is king or queen
(d) The card is a diamond
(e) The card is king
(f) The card is either red or ace
(g) The card is black
(h) The card is a 4 or lower

Solution:
The total of playing cards = 52
(a) The card is not king
Number of kings = 4
Probability of getting king = \(\frac { Number of Kings }{ Total number of cards } \)
= \(\frac { 4 }{ 52 } \) = \(\frac { 1 }{ 13 } \)
Probability of the card is not king = 1 – Probability of getting king
= 1 – \(\frac { 1 }{ 13 } \) = \(\frac { 12 }{ 13 } \)
(b) The card is red king
Probability of the card is a red king = \(\frac { Number of Red Kings }{ Total number of cards } \)
= \(\frac { 2 }{ 52 } \) = \(\frac { 1 }{ 26 } \)
(c) The card is king or queen
Number of kings = 4
Number of queens = 4
Total number of kings or queens = 4 + 4 = 8
P(card is a king or queen) = \(\frac { Number of Kings or Queens }{ Total number of cards } \)
= \(\frac { 8 }{ 52 } \) = \(\frac { 2 }{ 13 } \)
(d) The card is a diamond
Number of diamonds = 13
P(card is a diamond) = \(\frac { Number of diamonds }{ Total number of cards } \)
= \(\frac { 13 }{ 52 } \) = \(\frac { 1 }{ 4 } \)
(e) The card is king
Number of kings = 4
P(King) = \(\frac { Number of Kings }{ Total number of cards } \)
= \(\frac { 4 }{ 52 } \) = \(\frac { 1 }{ 13 } \)
(f) The card is either red or ace
Total number of red cards or ace cards = 28
P(card is either red card or ace) = \(\frac { Number of either cae or red cards }{ Total number of cards } \)
= \(\frac { 28 }{ 52 } \) = \(\frac { 7 }{ 13 } \)
(g) The card is black
No. of black cards = 26
P(card is black) = \(\frac { Number of black cards }{ Total number of cards } \)
= \(\frac { 26 }{ 52 } \) = \(\frac { 1 }{ 2 } \)
(h) The card is a 4 or lower
Number of cards is a 4 or lower = 12
P(card is a 4 or lower) = \(\frac { Number of 4 or lower cards }{ Total number of cards } \)
= \(\frac { 12 }{ 52 } \) = \(\frac { 3 }{ 13 } \)

Problem 4:
3 fair coins are tossed. What is the probability of getting at least 2 heads?

Solution:
Sample space S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Total no of outcomes = 8
Let E be the event of getting at least 2 heads = {HHT, HTH, THH, HHH}
Number of favorable outcomes = 4
P(E) = \(\frac { Number of favorable outcomes }{ Total number of outcomes } \)
= \(\frac { 4 }{ 8 } \)
= \(\frac { 1 }{ 2 } \)
Therefore, the probability of getting at least 2 heads while tossing 3 coins is \(\frac { 1 }{ 2 } \).

Problem 5:
Two dice are rolled simultaneously. Find the probability of (i) doublet (ii) product of 10 (iii) sum of at least 10?

Solution:
The total number of outcomes = 36
(i) Doublet
The possible doublets are {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)} = 6
P(doublets) = \(\frac { Number of favorable outcomes }{ Total number of outcomes } \)
= \(\frac { 6 }{ 36 } \)
= \(\frac { 1 }{ 6 } \)
(ii) Product of 10
The possible product of 10 are {(2,5), (5,2)} = 2
P(product of 10) = \(\frac { Number of favorable outcomes }{ Total number of outcomes } \)
= \(\frac { 2 }{ 36 } \)
= \(\frac { 1 }{ 18 } \)
(iii) Sum of at least 10
Possibilities of sum of 10 are {(4,6), (5,5), (5,6), (6,4), (6,5), (6,6)} = 6
P(sum of at lesat 10) = \(\frac { Number of favorable outcomes }{ Total number of outcomes } \)
= \(\frac { 6 }{ 36 } \)
= \(\frac { 1 }{ 6 } \)

Problem 6:
What is the probability of a die showing a number 2 or 6?

Solution:
P(2) is the probability of die showing 2
P(6) is the probability of die showing 6
P(2) = \(\frac { 1 }{ 6 } \), P(6) = \(\frac { 1 }{ 6 } \)
P(2 or 6) = P(2) + P(6)
= \(\frac { 1 }{ 6 } \) + \(\frac { 1 }{ 6 } \)
= \(\frac { 1 }{ 3 } \)